Prime Numbers in Arithmetic Progressions:
Dirichlet’s theorem and more
Sponsored by 
Lecturers
Associate Professor Timothy Trudgian, UNSW Canberra
Valeriia Starichkova, UNSW Canberra
Dr Michaela Cully-Hugill, UNSW Canberra
Synopsis
We will cover a several topics in Number Theory, the proof of the Dirichlet theorem and discussions around primes in arithmetic progressions. Participants will develop their problem-solving skills through various exercises and motivating discussions during the workshops. Particpiants will get acquainted with some of the basic techniques used nowadays in analytic number theory.
Course overview
Week 1:
- Elementary proofs about primes in arithmetic progressions
- Introduction to primes, primes congruent to 1 modulo and integer k
- Elementary attempts to prove the Dirichlet theorem via methods by Chebyshev and Erd˝os.
Week 2:
- Obstructions of elementary methods
- Introduction to complex analysis and analytic techniques covering the Riemann zeta-function and its extension to the complex plane using the functional equation
- An analytic proof of the infinitude of prime numbers.
Week 3:
- Introduction to the proof of the Dirichlet theorem: the Dirichlet characters and the properties of the Dirichlet L-functions.
Week 4:
- Finalising the proof of the Dirichlet theorem
- More questions around primes represented by polynomials and primes in arithmetic progressions like the Green-Tao theorem
Prerequisites
- Arithmetic progressions; acquaintance with modular arithmetic; complex numbers; one-variable calculus
- A basic understanding of complex analysis would be helpful but not required.
Assessment
TBA
Attendance requirements
Participation in all lectures and tutorials is expected.
For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.
Resources/pre-reading
TBA
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Associate Professor Timothy Trudgian, UNSW Canberra
Dr Timothy Trudgian is an associate professor at UNSW Canberra. Originally from Brisbane, he obtained a DPhil in mathematics from Oxford and returned to Canberra, after a two-year post-doc in Canada. His research interests lie in analytic number theory such as the Riemann zeta-function, distribution of primes, and primitive roots. He is a part of the Number Theory Group at UNSW Canberra and works closely with colleagues in number theory at UNSW Sydney
Dr Michaela Cully-Hugill, UNSW Canberra
Michaela Cully-Hugill is a recent PhD graduate in Number Theory from the University of New South Wales in Canberra. Her research has mainly been on explicit estimates for the distribution of primes, including error estimates for the Prime Number Theorem, and interval estimates for primes. She currently teaches at UNSW Canberra
Valeriia Starichkova, UNSW Canberra
Valeriia Starichkova is finishing her PhD in number theory at UNSW Canberra in September 2023. Her research interests lie in prime numbers in short intervals, sieve theory, the analytic properties of the Riemann zeta-function and the Dirichlet L-functions, and the connections to other areas such as probabilistic or algebraic number theory.